“The data were generated from random numbers, and there is no relation between X and Y at all. Firstly, values of X and Y were generated for each ‘subject,’ then a further random number was added to make the individual observation.”
From Bland and Altman, BMJ, 1994, 308, 896.
So… we follow their procedure.
##
## Call:
## lm(formula = y.new ~ x.new, data = individual_dataset)
##
## Residuals:
## Min 1Q Median 3Q Max
## -48.787 -7.580 4.742 14.604 33.979
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.8731 14.8849 -0.327 0.746339
## x.new 0.9154 0.2345 3.904 0.000713 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 25.12 on 23 degrees of freedom
## Multiple R-squared: 0.3986, Adjusted R-squared: 0.3724
## F-statistic: 15.24 on 1 and 23 DF, p-value: 0.0007132## Linear mixed model fit by REML ['lmerMod']
## Formula: y.new ~ x.new + (1 | groupnum)
## Data: individual_dataset
##
## REML criterion at convergence: 182.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.3098 -0.6503 -0.3808 0.6974 1.5197
##
## Random effects:
## Groups Name Variance Std.Dev.
## groupnum (Intercept) 858.47 29.300
## Residual 46.83 6.843
## Number of obs: 25, groups: groupnum, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 27.2568 17.7559 1.535
## x.new 0.3777 0.1992 1.896
##
## Correlation of Fixed Effects:
## (Intr)
## x.new -0.670